A Scholar's class defining feature isn't the shielding that is attached to two of their heals--it's the faerie. Shields are part of the Scholar toolkit, just as Regen is part of the White Mage's toolkit. This isn't WoW where a Priest has tools like Spirit Shell or Power Word: Shield. A Scholar's role is healing, with shielding being part of that role.

Which is better? That depends a great deal on the situation. In a raid, having both a SCH and a WHM is preferable, with each bringing different tools to the table. A White Mage has an easy time dealing with huge raidwide damage, while a Scholar is better able to support a tank who is getting hit for 2-3k every few seconds. You can certainly healing through AoE damage as a Scholar, just as a White Mage can keep up a tank taking heavy damage, but it will be harder and less MP efficient for you to do so.

In short, stop bickering. Both are main healers who work very well together.